1.1 MCQs-Sets

Sets

Basic Concepts and Definitions

1. A set is defined as:

1. A collection of similar objects

2. A well-defined collection of distinct objects

3. A group of numbers

4. Any collection of things

Show me the answer

Answer: 2. A well-defined collection of distinct objects

Explanation:

• A set must be well-defined, meaning it's clear whether any given object is a member of the set or not.

• The objects (elements) in a set must be distinct (no repetitions).

• The order of elements does not matter in a set.

• Example: $\{1, 2, 3\}$ and $\{3, 1, 2\}$ represent the same set.

2. Which of the following is NOT a correct way to describe a set?

1. $\{1, 3, 5, 7\}$

2. $\{x \mid x \text{ is a prime number less than 10}\}$

3. $\{2, 4, 6, 2, 8\}$

4. $\{a, e, i, o, u\}$

Show me the answer

Answer: 3. $\{2, 4, 6, 2, 8\}$

Explanation:

• In a set, elements should be distinct and not repeated.

• Option 3 has the element '2' repeated, which violates the definition of a set.

• Option 1 uses the roster method (listing all elements).

• Option 2 uses the set-builder notation.

• Option 4 is the set of vowels in English.

3. The set $\{x : x \in \mathbb{N} \text{ and } x < 1\}$ is:

1. $\{0\}$

2. $\{1\}$

3. $\emptyset$

4. $\{0, 1\}$

Show me the answer

Answer: 3. $\emptyset$

Explanation:

• The set contains all natural numbers ($\mathbb{N}$) that are less than 1.

• By convention, $\mathbb{N} = \{1, 2, 3, \ldots\}$ (natural numbers start from 1).

• There is no natural number less than 1.

• Therefore, the set is empty, denoted by $\emptyset$ or $\{\}$.

4. Which of the following represents a singleton set?

1. $\{a, b, c\}$

2. $\{5\}$

3. $\{\emptyset\}$

4. Both 2 and 3

Show me the answer

Answer: 4. Both 2 and 3

Explanation:

• A singleton set contains exactly one element.

• $\{5\}$ has one element: the number 5.

• $\{\emptyset\}$ has one element: the empty set $\emptyset$ itself.

• It's important to distinguish $\emptyset$ (empty set, 0 elements) from $\{\emptyset\}$ (singleton set, 1 element).

• $\{a, b, c\}$ has three elements, so it's not a singleton.

5. Two sets A and B are said to be equal if:

1. They have the same number of elements

2. Every element of A is in B and every element of B is in A

3. They are both subsets of each other

4. Both 2 and 3

Show me the answer

Answer: 4. Both 2 and 3

Explanation:

• Sets A and B are equal ($A = B$) if they contain exactly the same elements.

• Formally: $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$.

• Option 2 is the direct definition: every element of A is in B, and every element of B is in A.

• Option 3 is equivalent to option 2 through the definition of subset.

• Option 1 describes equivalent sets (same cardinality), not necessarily equal sets.

Subsets and Power Sets

6. If a set A has 5 elements, how many subsets does it have?

1. 5

2. 10

3. 25

4. 32

Show me the answer

Answer: 4. 32

Explanation:

• The number of subsets of a set with n elements is $2^n$.

• Here, n = 5, so number of subsets = $2^5 = 32$.

• This includes:

  • The empty set: 1

  • Singleton subsets: 5

  • 2-element subsets: $\binom{5}{2} = 10$

  • 3-element subsets: $\binom{5}{3} = 10$

  • 4-element subsets: $\binom{5}{4} = 5$

  • The set itself: 1

• Total = 1 + 5 + 10 + 10 + 5 + 1 = 32.

7. For any set A, which of the following is always TRUE?

1. $A \in A$

2. $\emptyset \in A$

3. $\emptyset \subseteq A$

4. $A \subset A$

Show me the answer

Answer: 3. $\emptyset \subseteq A$

Explanation:

• The empty set $\emptyset$ is a subset of every set, including itself.

• Formally: $\emptyset \subseteq A$ for any set A.

• $A \in A$ is generally false (except in some axiomatic set theories with non-well-founded sets).

• $\emptyset \in A$ is only true if A explicitly contains the empty set as an element.

• $A \subset A$ is false because $A \subseteq A$ (A is a subset of itself), but not a proper subset.

8. The power set of $\{a, b\}$ is:

1. $\{\{a\}, \{b\}\}$

2. $\{\emptyset, \{a\}, \{b\}, \{a, b\}\}$

3. $\{\emptyset, a, b, \{a, b\}\}$

4. $\{a, b, \{a\}, \{b\}\}$

Show me the answer

Answer: 2. $\{\emptyset, \{a\}, \{b\}, \{a, b\}\}$

Explanation:

• The power set of a set A, denoted $P(A)$, is the set of all subsets of A.

• For A = $\{a, b\}$, the subsets are:

  • Empty set: $\emptyset$

  • Singleton subsets: $\{a\}$, $\{b\}$

  • The set itself: $\{a, b\}$

• Therefore, $P(\{a, b\}) = \{\emptyset, \{a\}, \{b\}, \{a, b\}\}$.

• Note: The elements of the power set are sets themselves.

9. If $A \subseteq B$ and $B \subseteq A$, then:

1. A and B are disjoint

2. A is a proper subset of B

3. A = B

4. B is a proper subset of A

Show me the answer

Answer: 3. A = B

Explanation:

• If $A \subseteq B$, then every element of A is in B.

• If $B \subseteq A$, then every element of B is in A.

• Together, these imply that A and B have exactly the same elements.

• Therefore, $A = B$.

• This is known as the antisymmetry property of set inclusion.

10. How many proper subsets does a set with 4 elements have?

1. 15

2. 16

3. 4

4. 8

Show me the answer

Answer: 1. 15

Explanation:

• For a set with n elements:

  • Total number of subsets = $2^n$

  • Number of proper subsets = $2^n - 1$ (excluding the set itself)

• Here, n = 4:

  • Total subsets = $2^4 = 16$

  • Proper subsets = 16 - 1 = 15

• Proper subsets include all subsets except the original set itself.

Universal Set and Complement

11. If U is the universal set and A is any subset of U, then $A \cup A'$ equals:

1. $\emptyset$

2. A

3. $A'$

4. U

Show me the answer

Answer: 4. U

Explanation:

• $A'$ (complement of A) contains all elements in U that are not in A.

• The union $A \cup A'$ contains all elements that are either in A OR in A' (or both).

• Since every element of U is either in A or in A' (by definition of complement), the union equals the entire universal set.

• This is one of the complement laws: $A \cup A' = U$.

12. For any set A, $(A')'$ equals:

1. $\emptyset$

2. U

3. A

4. $A'$

Show me the answer

Answer: 3. A

Explanation:

• This is the involution law or double complement law.

• The complement of A, denoted $A'$, contains all elements not in A.

• The complement of $A'$, denoted $(A')'$, contains all elements not in $A'$.

• An element not in $A'$ must be in A.

• Therefore, $(A')' = A$.

• Example: If U = {1,2,3,4,5} and A = {1,2}, then A' = {3,4,5}, and (A')' = {1,2} = A.

13. Which of the following is NOT true for any sets A and B?

1. $(A \cup B)' = A' \cap B'$

2. $(A \cap B)' = A' \cup B'$

3. $A - B = A \cap B'$

4. $A - B = B - A$

Show me the answer

Answer: 4. $A - B = B - A$

Explanation:

• Options 1 and 2 are De Morgan's Laws.

• Option 3 is true: $A - B$ (elements in A but not in B) equals $A \cap B'$.

• Option 4 is generally false: $A - B$ and $B - A$ are different sets unless A = B.

• Example: Let A = {1,2,3}, B = {3,4,5}

  • $A - B = \{1,2\}$

  • $B - A = \{4,5\}$

  • Clearly, $\{1,2\} \neq \{4,5\}$

14. If U = {1,2,3,4,5,6,7,8,9,10}, A = {2,4,6,8,10}, then $A'$ is:

1. {1,3,5,7,9}

2. {2,4,6,8,10}

3. {1,2,3,4,5,6,7,8,9,10}

4. $\emptyset$

Show me the answer

Answer: 1. {1,3,5,7,9}

Explanation:

• The complement $A'$ contains all elements in U that are not in A.

• U = {1,2,3,4,5,6,7,8,9,10}

• A = {2,4,6,8,10} (even numbers from 1 to 10)

• Elements not in A: {1,3,5,7,9} (odd numbers from 1 to 10)

• Therefore, $A' = \{1,3,5,7,9\}$.

Operations on Sets

15. For sets A = {1,2,3,4} and B = {3,4,5,6}, $A \cup B$ is:

1. {3,4}

2. {1,2,5,6}

3. {1,2,3,4,5,6}

4. {1,2,3,4}

Show me the answer

Answer: 3. {1,2,3,4,5,6}

Explanation:

• The union $A \cup B$ contains all elements that are in A OR in B (or both).

• A = {1,2,3,4}

• B = {3,4,5,6}

• Elements in A: 1,2,3,4

• Elements in B: 3,4,5,6

• Combined unique elements: 1,2,3,4,5,6

• Therefore, $A \cup B = \{1,2,3,4,5,6\}$.

16. For sets A = {1,2,3,4} and B = {3,4,5,6}, $A \cap B$ is:

1. {3,4}

2. {1,2,5,6}

3. {1,2,3,4,5,6}

4. {1,2,3,4}

Show me the answer

Answer: 1. {3,4}

Explanation:

• The intersection $A \cap B$ contains all elements that are in BOTH A AND B.

• A = {1,2,3,4}

• B = {3,4,5,6}

• Common elements: 3 and 4

• Therefore, $A \cap B = \{3,4\}$.

17. For sets A = {1,2,3,4} and B = {3,4,5,6}, $A - B$ is:

1. {3,4}

2. {1,2}

3. {5,6}

4. {1,2,5,6}

Show me the answer

Answer: 2. {1,2}

Explanation:

• The difference $A - B$ contains elements that are in A but NOT in B.

• A = {1,2,3,4}

• B = {3,4,5,6}

• Elements in A: 1,2,3,4

• Remove elements that are also in B: 3,4

• Remaining: 1,2

• Therefore, $A - B = \{1,2\}$.

18. For sets A = {1,2,3,4} and B = {3,4,5,6}, the symmetric difference $A \triangle B$ is:

1. {3,4}

2. {1,2,5,6}

3. {1,2,3,4,5,6}

4. {1,2}

Show me the answer

Answer: 2. {1,2,5,6}

Explanation:

• Symmetric difference $A \triangle B$ contains elements that are in either A or B, but NOT in both.

• Definition: $A \triangle B = (A - B) \cup (B - A)$

• From previous questions:

  • $A - B = \{1,2\}$

  • $B - A = \{5,6\}$

• Therefore, $A \triangle B = \{1,2\} \cup \{5,6\} = \{1,2,5,6\}$.

• Alternatively: $A \triangle B = (A \cup B) - (A \cap B) = \{1,2,3,4,5,6\} - \{3,4\} = \{1,2,5,6\}$.

Laws of Set Algebra

19. Which law states that $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$?

1. Commutative law

2. Associative law

3. Distributive law

4. De Morgan's law

Show me the answer

Answer: 3. Distributive law

Explanation:

• This is the distributive law of union over intersection.

• Just like in algebra: $a \times (b + c) = (a \times b) + (a \times c)$

• In set theory: union distributes over intersection.

• There's also the dual distributive law: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

• This law can be proven using element arguments or Venn diagrams.

20. De Morgan's Law states that:

1. $(A \cup B)' = A' \cap B'$

2. $(A \cap B)' = A' \cup B'$

3. Both 1 and 2

4. $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

Show me the answer

Answer: 3. Both 1 and 2

Explanation:

• De Morgan's Laws are two important laws in set theory:

  1. The complement of a union equals the intersection of complements: $(A \cup B)' = A' \cap B'$

  2. The complement of an intersection equals the union of complements: $(A \cap B)' = A' \cup B'$

• These laws also apply in logic (for AND/OR operations) and Boolean algebra.

• They can be proven using element arguments or Venn diagrams.

21. The commutative law for sets states that:

1. $A \cup B = B \cup A$

2. $A \cap B = B \cap A$

3. Both 1 and 2

4. $A \cup (B \cup C) = (A \cup B) \cup C$

Show me the answer

Answer: 3. Both 1 and 2

Explanation:

• Commutative laws state that the order of operands doesn't matter for union and intersection:

  • $A \cup B = B \cup A$

  • $A \cap B = B \cap A$

• However, note that:

  • $A - B \neq B - A$ (difference is not commutative)

  • $A \times B \neq B \times A$ (Cartesian product is not commutative unless A = B)

• Option 4 represents the associative law for union.

22. Which of the following represents the absorption law?

1. $A \cup (A \cap B) = A$

2. $A \cap (A \cup B) = A$

3. Both 1 and 2

4. $A \cup A = A$

Show me the answer

Answer: 3. Both 1 and 2

Explanation:

• Absorption laws state that:

  1. $A \cup (A \cap B) = A$

  2. $A \cap (A \cup B) = A$

• These laws describe how a set "absorbs" certain combinations with itself.

• They can be verified using Venn diagrams or logical reasoning:

  • For (1): Any element in $A \cup (A \cap B)$ is either in A or in both A and B, so it must be in A.

  • For (2): Any element in $A \cap (A \cup B)$ is in A and also in either A or B, so it must be in A.

• Option 4 represents the idempotent law.

Cardinality of Sets

23. For any two finite sets A and B, the formula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ is known as:

1. De Morgan's principle

2. Inclusion-Exclusion principle

3. Distributive principle

4. Complement principle

Show me the answer

Answer: 2. Inclusion-Exclusion principle

Explanation:

• The Inclusion-Exclusion principle for two sets states: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

• Reason: When we add $n(A)$ and $n(B)$, elements in $A \cap B$ are counted twice.

• So we subtract $n(A \cap B)$ once to count them only once.

• For three sets: $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$

24. If $n(A) = 20$, $n(B) = 30$, and $n(A \cap B) = 10$, then $n(A \cup B)$ is:

1. 40

2. 50

3. 60

4. 70

Show me the answer

Answer: 1. 40

Explanation:

• Using the Inclusion-Exclusion principle: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

• Substitute given values: $n(A \cup B) = 20 + 30 - 10 = 40$

• Verification:

  • Elements only in A: $n(A) - n(A \cap B) = 20 - 10 = 10$

  • Elements only in B: $n(B) - n(A \cap B) = 30 - 10 = 20$

  • Elements in both: $n(A \cap B) = 10$

  • Total: 10 + 20 + 10 = 40

25. If $n(A) = 50$, $n(B) = 40$, and $n(A \cup B) = 70$, then $n(A \cap B)$ is:

1. 10

2. 20

3. 30

4. 40

Show me the answer

Answer: 2. 20

Explanation:

• From Inclusion-Exclusion: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

• Rearranging: $n(A \cap B) = n(A) + n(B) - n(A \cup B)$

• Substitute: $n(A \cap B) = 50 + 40 - 70 = 20$

• This means there are 20 elements common to both sets A and B.

26. In a class of 60 students, 35 play cricket and 25 play football. If 10 play both, how many play neither?

1. 10

2. 15

3. 20

4. 25

Show me the answer

Answer: 1. 10

Explanation:

• Let C = set of cricket players, F = set of football players

• Given: $n(U) = 60$, $n(C) = 35$, $n(F) = 25$, $n(C \cap F) = 10$

• Using Inclusion-Exclusion: $n(C \cup F) = 35 + 25 - 10 = 50$

• This means 50 students play at least one sport.

• Students playing neither = Total - Students playing at least one $= 60 - 50 = 10$

Cartesian Product

27. The Cartesian product $A \times B$ is defined as:

1. $\{a, b : a \in A, b \in B\}$

2. $\{(a, b) : a \in A, b \in B\}$

3. $\{b, a : b \in B, a \in A\}$

4. $(a, b)$ for all $a \in A, b \in B$

Show me the answer

Answer: 2. $\{(a, b) : a \in A, b \in B\}$

Explanation:

• The Cartesian product $A \times B$ is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

• Important properties:

  • $(a, b)$ is an ordered pair: $(a, b) \neq (b, a)$ unless a = b

  • $A \times B \neq B \times A$ in general

• Example: If A = {1,2}, B = {a,b}, then: $A \times B = \{(1,a), (1,b), (2,a), (2,b)\}$ $B \times A = \{(a,1), (a,2), (b,1), (b,2)\}$

28. If A has 3 elements and B has 4 elements, then $A \times B$ has:

1. 7 elements

2. 12 elements

3. 1 element

4. Cannot be determined

Show me the answer

Answer: 2. 12 elements

Explanation:

• The number of elements in $A \times B$ is the product of the number of elements in A and B.

• Formula: $n(A \times B) = n(A) \times n(B)$

• Here: $n(A) = 3$, $n(B) = 4$

• Therefore: $n(A \times B) = 3 \times 4 = 12$

• Each element of A can be paired with each element of B.

29. For any set A, $A \times \emptyset$ equals:

1. A

2. $\emptyset$

3. $\{(a, \emptyset) : a \in A\}$

4. $\{(\emptyset, a) : a \in A\}$

Show me the answer

Answer: 2. $\emptyset$

Explanation:

• The Cartesian product with the empty set is always empty.

• Reason: For $A \times \emptyset$ to have an element $(a, b)$, we need $a \in A$ and $b \in \emptyset$.

• But there is no $b \in \emptyset$ (empty set has no elements).

• Therefore, $A \times \emptyset = \emptyset$.

• Similarly, $\emptyset \times A = \emptyset$.

30. Which distributive property is TRUE for Cartesian product?

1. $A \times (B \cup C) = (A \times B) \cup (A \times C)$

2. $A \times (B \cap C) = (A \times B) \cap (A \times C)$

3. Both 1 and 2

4. Neither 1 nor 2

Show me the answer

Answer: 3. Both 1 and 2

Explanation:

• Cartesian product distributes over both union and intersection:

  1. $A \times (B \cup C) = (A \times B) \cup (A \times C)$

  2. $A \times (B \cap C) = (A \times B) \cap (A \times C)$

• Also distributes over set difference: $A \times (B - C) = (A \times B) - (A \times C)$

• These can be proven using element arguments: For (1): $(x, y) \in A \times (B \cup C) \Leftrightarrow x \in A \text{ and } y \in (B \cup C)$ $\Leftrightarrow x \in A \text{ and } (y \in B \text{ or } y \in C)$ $\Leftrightarrow (x \in A \text{ and } y \in B) \text{ or } (x \in A \text{ and } y \in C)$ $\Leftrightarrow (x, y) \in (A \times B) \text{ or } (x, y) \in (A \times C)$ $\Leftrightarrow (x, y) \in (A \times B) \cup (A \times C)$